Proof: Let \(x \in \mathbb {X}\)  and \(\psi \in {\mathbf {N}}^*\) .
By the multivariate Mecke equation ([1]}),
\(\kappa (x,&\psi _x) Z_{(x,\infty )}(\psi _x+\delta _x)=\kappa (x,\psi _x) \int e^{-H(\mu ,\psi _x+\delta _x)}\,\mathrm {d}\Pi _{\lambda _{(x,\infty )}}(\mathrm {d}\mu )\\&=\kappa (x,\psi _x) e^{-\lambda ((x,\infty ))}\left(1+\sum _{m=1}^\infty \frac{1}{m!}\int _{\mathbb {X}^m} \kappa _m(x_1,\dots ,x_m,\psi _x+\delta _x)\,\lambda _{(x,\infty )}^m(\mathrm {d}(x_1,\dots ,x_m))\right)\\&=e^{-\lambda ((x,\infty ))}\left(\kappa (x,\psi _x) + \sum _{m=1}^\infty \frac{1}{m!}\int _{\mathbb {X}^m} \kappa _{m+1}(x_1,\dots ,x_m,x,\psi _x)\,\lambda _{(x,\infty )}^m(\mathrm {d}(x_1,\dots ,x_m))\right).\) 
